Absolutely great game. Fans of Batman will love the vast array of cameos and easter-eggs. There are plenty of bad guys to fight and super-villians to deal with. The Catwoman DLC/Add-on is a little over priced for what you get and the Robin pack is pretty worthless and adds no in-game content, he is just featured in the challenges section. The challenges section I thought was kind of weak and repetitive but great for fight fans. Overall though this game was awesome even if you aren't a huge fan of Batman and just want to beat up bad guys!
Batman: Arkham City is a masterpiece. That's the main thing you need to know. Arkham Asylum pails in comparison (hope I'm using that phrase right :D). Now onto the review.
Story: Batman:AC takes place 2 years after the events of Arkham Asylum. The warden in the last game, Quincy Sharp is now the mayor of Gotham. Hugo Strange (The main villain in this one) convinces Sharp that it would be a good idea to section off part of Gotham City and throw every criminal and lowlife inside and make it their new home. Bruce Wayne gets thrown in there to but you have to play the game to figure out how and why ;).
Graphics: As far as graphics go Rocksteady blew it out of the park. Batman:AC uses the unreal 3 engine, and it definitely has a shot as the best looking game this year.
World: Batman: AC is an open world game, unlike Arkham Asylum, in this one you have a city to explore. There's a lot to look at and while it isn't the biggest open world game by any means, it's filled to the brim with swathes of detail. There isn't a single area that looks empty. Rocksteady painstakingly filled the entire city with references and love.
Voice Acting: It's top notch, with arguably the best performances so far this year. Kevin Conroy (Batman) Mark Hamil (The Joker) and Nolan North (The Penguin) Definitely stand out of the crowd.
Replay-Ability: One of my favorite things about Arkham Asylum was the collectables, they weren't just something to extend the game life they were a boss battle with the Riddler. They were challenging to find and unlocked cool extras. AC is no different stuffed full of over 400 collectables and riddles (and more!) That's nearly double of the first game. Plus the added edition of a new game plus mode where you start the game off at the level you were when you beat the game normally, you get to keep all your gadgets and abilities, but the enemies are a lot tougher, means AC has a lot going for it.
Final Verdict: 10/10 If your a Batman fan, you'll ador